Theorem ffnfv 2892

Description: A function maps to a class to which all values belong.

Assertion

Ref	Expression
ffnfv	|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Distinct variable group(s):   x,A   x,B   x,F

Proof of Theorem ffnfv

Step	Hyp	Ref	Expression
1		ffn 2752	. . 3 |- (F:A-->B -> F Fn A)
2		ffvrn 2890	. . . . 5 |- ((F:A-->B /\ x e. A) -> (F` x) e. B)
3	2	exp 291	. . . 4 |- (F:A-->B -> (x e. A -> (F` x) e. B))
4	3	r19.21aiv 1259	. . 3 |- (F:A-->B -> A.x e. A (F` x) e. B)
5	1, 4	jca 236	. 2 |- (F:A-->B -> (F Fn A /\ A.x e. A (F` x) e. B))
6		pm3.26 256	. . . 4 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> F Fn A)
7		fvelrn 2883	. . . . . . . 8 |- (F Fn A -> (y e. ran F <-> E.x e. A (F` x) = y))
8	7	biimpd 135	. . . . . . 7 |- (F Fn A -> (y e. ran F -> E.x e. A (F` x) = y))
9		hbra1 1237	. . . . . . . 8 |- (A.x e. A (F` x) e. B -> A.xA.x e. A (F` x) e. B)
10		ax-17 925	. . . . . . . 8 |- (y e. B -> A.x y e. B)
11		ra4 1243	. . . . . . . . 9 |- (A.x e. A (F` x) e. B -> (x e. A -> (F` x) e. B))
12		eleq1 1149	. . . . . . . . . 10 |- ((F` x) = y -> ((F` x) e. B <-> y e. B))
13	12	biimpcd 137	. . . . . . . . 9 |- ((F` x) e. B -> ((F` x) = y -> y e. B))
14	11, 13	syl6 23	. . . . . . . 8 |- (A.x e. A (F` x) e. B -> (x e. A -> ((F` x) = y -> y e. B)))
15	9, 10, 14	r19.23ad 1285	. . . . . . 7 |- (A.x e. A (F` x) e. B -> (E.x e. A (F` x) = y -> y e. B))
16	8, 15	sylan9 359	. . . . . 6 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> (y e. ran F -> y e. B))
17	16	r19.21aiv 1259	. . . . 5 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> A.y e. ran Fy e. B)
18		dfss3 1498	. . . . 5 |- (ran F (_ B <-> A.y e. ran Fy e. B)
19	17, 18	sylibr 175	. . . 4 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> ran F (_ B)
20	6, 19	jca 236	. . 3 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> (F Fn A /\ ran F (_ B))
21		df-f 2434	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
22	20, 21	sylibr 175	. 2 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> F:A-->B)
23	5, 22	impbi 139	1 |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   (_ wss 1487  ran crn 2411   Fn wfn 2417  -->wf 2418  ` cfv 2422

This theorem is referenced by:  fnfvrnss 2893  fopabfv 2894  abianfp 3000  ffnoprval 3041  mapxpen 3390  unblem4 3434

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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